The harmonic 1-form \(\xi \in \mathrm {Harm}^1(M)\)  depends non-trivially
on \(\mathbf {a}:=\left(a_1, \, \ldots , \, a_n\right)\) 
and \(\mathbf {d}:=(d_1, \, \ldots , \, d_n)\) , as it must satisfy
the constraint \(\xi \in \mathcal {L}(\mathbf {a}, \, \mathbf {d})\) 
where \(\mathcal {L}(\mathbf {a}, \, \mathbf {d})\)  is a suitable subset of \(\mathrm {Harm}^1(M)\) 
defined by the position and the charges of the singularities.
(See [1]} and Section 
for the definition of \(\mathcal {L}(\mathbf {a}, \, \mathbf {d})\) .)
If the \(u_\varepsilon \) 's are minimizers of \(F_\varepsilon \) , then \(n = \left| \chi (M) \right|\) ,
\(d_j = \operatorname{sign}(\chi (M))\)  for any \(j\)  and
\( F_\varepsilon (u_\varepsilon ) = \pi n\left| \log \varepsilon  \right|+ W(\mathbf {a}, \, \mathbf {d}, \, \xi ) + n\gamma + \mathrm {o}(1)\qquad \textrm {as }\varepsilon \rightarrow 0.\) 
